The work developed in this thesis focusses on the electrical activity of the heart, from the modeling of the action potential originating from cardiac cells and propagating through the heart, as well as its electrical manifestation at the body surface. The study is divided in two main parts: modeling the action potential, and numerical simulations. For modeling the action potential a dimensional and asymptotic analysis is done. The key advance in this part of the work is that this analysis gives the steps to reliably control the action potential. It allows predicting the time/space scales and speed of any action potential that is to say the shape of the action potential and its propagation. This can be done as the explicit relations on all the physiological constants are defined precisely. This method facilitates the integrative modeling of a complete human heart with tissue-specific ionic models. It even proves that using a single model for the cardiac action potential is enough in many situations. For efficient numerical simulations, a numerical method for solving the heart-torso coupling problem is explored according to a level set description of the domains. This is done in the perspective of using directly medical images for building computational domains. A finite element method is then developed to manage meshes not adapted to internal interfaces. Finally, an anisotropic adaptive remeshing methods for unstructured finite element meshes is used to efficiently capture propagating action potentials within complex, realistic two dimensional geometries.

The work developed in this thesis focusses on the electrical activity of the heart, from the modeling of the action potential originating from cardiac cells and propagating through the heart, as well as its electrical manifestation at the body surface. The study is divided in two main parts: modeling the action potential, and numerical simulations. For modeling the action potential a dimensional and asymptotic analysis is done. The key advance in this part of the work is that this analysis gives the steps to reliably control the action potential. It allows predicting the time/space scales and speed of any action potential that is to say the shape of the action potential and its propagation. This can be done as the explicit relations on all the physiological constants are defined precisely. This method facilitates the integrative modeling of a complete human heart with tissue-specific ionic models. It even proves that using a single model for the cardiac action potential is enough in many situations. For efficient numerical simulations, a numerical method for solving the heart-torso coupling problem is explored according to a level set description of the domains. This is done in the perspective of using directly medical images for building computational domains. A finite element method is then developed to manage meshes not adapted to internal interfaces. Finally, an anisotropic adaptive remeshing methods for unstructured finite element meshes is used to efficiently capture propagating action potentials within complex, realistic two dimensional geometries.

The work developed in this thesis focusses on the electrical activity of the heart, from the modeling of the action potential originating from cardiac cells and propagating through the heart, as well as its electrical manifestation at the body surface. The study is divided in two main parts: modeling the action potential, and numerical simulations. For modeling the action potential a dimensional and asymptotic analysis is done. The key advance in this part of the work is that this analysis gives the steps to reliably control the action potential. It allows predicting the time/space scales and speed of any action potential that is to say the shape of the action potential and its propagation. This can be done as the explicit relations on all the physiological constants are defined precisely. This method facilitates the integrative modeling of a complete human heart with tissue-specific ionic models. It even proves that using a single model for the cardiac action potential is enough in many situations. For efficient numerical simulations, a numerical method for solving the heart-torso coupling problem is explored according to a level set description of the domains. This is done in the perspective of using directly medical images for building computational domains. A finite element method is then developed to manage meshes not adapted to internal interfaces. Finally, an anisotropic adaptive remeshing methods for unstructured finite element meshes is used to efficiently capture propagating action potentials within complex, realistic two dimensional geometries.

The work developed in this thesis focusses on the electrical activity of the heart, from the modeling of the action potential originating from cardiac cells and propagating through the heart, as well as its electrical manifestation at the body surface. The study is divided in two main parts: modeling the action potential, and numerical simulations. For modeling the action potential a dimensional and asymptotic analysis is done. The key advance in this part of the work is that this analysis gives the steps to reliably control the action potential. It allows predicting the time/space scales and speed of any action potential that is to say the shape of the action potential and its propagation. This can be done as the explicit relations on all the physiological constants are defined precisely. This method facilitates the integrative modeling of a complete human heart with tissue-specific ionic models. It even proves that using a single model for the cardiac action potential is enough in many situations. For efficient numerical simulations, a numerical method for solving the heart-torso coupling problem is explored according to a level set description of the domains. This is done in the perspective of using directly medical images for building computational domains. A finite element method is then developed to manage meshes not adapted to internal interfaces. Finally, an anisotropic adaptive remeshing methods for unstructured finite element meshes is used to efficiently capture propagating action potentials within complex, realistic two dimensional geometries.

La modélisation du vivant, en particulier la modélisation de l'activité cardiaque, est devenue un défi scientifique majeur. Le but de cette thématique est de mieux comprendre les phénomènes physiologiques et donc d'apporter des solutions à des problèmes cliniques. Nous nous intéressons dans cette thèse à la modélisation et à l'étude numérique de l'activité électrique du cœur, en particulier l'étude des électrocardiogrammes (ECGs). L'onde électrique dans le cœur est gouvernée par un système d'équations de réaction-diffusion appelé modèle bidomaine ce système est couplé à une EDO représentant l'activité cellulaire. Afin simuler des ECGs, nous tenons en compte la propagation de l'onde électrique dans le thorax qui est décrite par une équation de diffusion. Nous commençons par une démonstrer l'existence d'une solution faible du système couplé cœur-thorax pour une classe de modèles ioniques phénoménologiques. Nous prouvons ensuite l'unicité de cette solution sous certaines conditions. Le plus grand apport de cette thèse est l'étude et la simulation numérique du couplage électrique cœur-thorax. Les résultats de simulations sont représentés à l'aide des ECGs. Dans une première partie, nous produisons des simulations pour un cas normal et pour des cas pathologiques (blocs de branche gauche et droit et des arhythmies). Nous étudions également l'impact de certaines hypothèses de modélisation sur les ECGs (couplage faible, utilisation du modèle monodomaine, isotropie, homogénéité cellulaire, comportement résistance-condensateur du péricarde,. . . ). Nous étudions à la fin de cette partie la sensibilité des ECGs par apport aux paramètres du modèle. En deuxième partie, nous effectuons l'analyse numérique de schémas du premier ordre en temps découplant les calculs du potentiel d'action et du potentiel extérieur. Puis, nous combinons ces schémas en temps avec un traîtement explicite du type Robin-Robin des conditions de couplage entre le cœur et le thorax. Nous proposons une analyse de stabilité de ces schémas et nous illustrons les résultats avec des simulations numériques d'ECGs. La dernière partie est consacrée à trois applications. Nous commençons par l'estimation de certains paramètres du modèle (conductivité du thorax et paramètres ioniques). Dans la deuxième application, qui est d'originie industrielle, nous utilisons des méthodes d'apprentissage statistique pour reconstruire des ECGs à partir de mesures ('électrogrammes). Enfin, nous présentons des simulations électro-mécaniques du coeur sur une géométrie réelle dans diverses situations physiologiques et pathologiques. Les indicateurs cliniques, électriques et mécaniques, calculés à partir de ces simulations sont très similaires à ceux observés en réalité.